Article
Full entry |
PDF
(0.4 MB)
Feedback
PDF
(0.4 MB)
Feedback
Keywords:
finite group; solvable group; order of element
finite group; solvable group; order of element
Summary:
Let $G$ be a finite group and $\operatorname{nse}(G)$ the set of numbers of elements with the same order in $G$. In this paper, we prove that a finite group $G$ is isomorphic to $M$, where $M$ is one of the Mathieu groups, if and only if the following hold:
(1) $|G|=|M|$,
(2) $\operatorname{nse}(G)=\operatorname{nse}(M)$.
References:
[1] Cao, Z. F.: $S(2,3,5,7,11)$ and simple $K_n$-group. J. Heilongjiang Univ. Natur. Sci. 15 (2) (1998), 1–5, in Chinese. MR 1671457
[2] Cao, Z. F.: Diophantine equation and it’s application. Shanghai Jiaotong University Press, 2000, in Chinese.
[3] Chillag, D., Herzog, M.: On the length of the conjugacy classes of finite groups. J. Algebra 131 (1) (1990), 110–125. DOI 10.1016/0021-8693(90)90168-N | MR 1055001 | Zbl 0694.20015
[4] Conway, J. H., Curtis, R. T., etc., S. P. Norton: Atlas of Finite Groups. Oxford, Clarendon Press, 1985. MR 0827219 | Zbl 0568.20001
[5] Cossey, J., Wang, Y.: Remarks on the length of conjugacy classes of finite groups. Comm. Algebra 27 (9) (1999), 4347–4353. DOI 10.1080/00927879908826701 | MR 1705872 | Zbl 0948.20010
[6] Hall, P.: A note on soluble groups. J. London Math. Soc. 3 (2) (1928), 98–105. DOI 10.1112/jlms/s1-3.2.98
[7] Herzog, M.: On finite simple groups of order divisible by three primes only. J. Algebra 120 (10) (1968), 383–388. DOI 10.1016/0021-8693(68)90088-4 | MR 0233881
[8] Ito, N.: Simple groups of conjugate type rank 4. J. Algebra 20 (1972), 226–249. DOI 10.1016/0021-8693(72)90057-9 | MR 0289636 | Zbl 0228.20004
[9] Jafarzadeh, A., Iranmanesh, A.: On simple $K_n$-groups for $n=5, 6$. London Math. Soc. Lecture Note Ser. (Campbell, C. M., Quick, M. R., Robertson, E. F., Smith, G. C., eds.), Cambridge University Press, 2007. Zbl 1115.20009
[10] Kurzweil, H., Stellmacher, B.: The Theory of Finite Groups. Springer-Verlag Berlin, 2004. MR 2014408 | Zbl 1047.20011
[11] Shi, W. J.: A new characterization of the sporadic simple groups. Group Theory, Proc. of the 1987 Singapore Conf., Walter de Gruyter, Berlin, 1989, pp. 531–540. MR 0981868 | Zbl 0657.20017
[12] Shi, W. J.: On simple $K_4$-groups. Chinese Sci. Bull. 36 (17) (1991), 1281–1283, in Chinese.
[13] Shi, W. J.: The quantitative structure of groups and related topics. Math. Appl. (China Ser.) 365 (1996), 163–181, Kluwer Acad. Publ., Dordrecht. MR 1447204 | Zbl 0872.20026
[14] Shi, W. J.: Pure quantitative characterization of finite simple groups. Front. Math. China 2 (1) (2007), 123–125. DOI 10.1007/s11464-007-0008-3 | MR 2289913 | Zbl 1198.20016
Search
Partner of
